Equations that are posed over a space of probability measures, or more general metric spaces, have been considered in a number of works including [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. These works revolve around equations involving derivatives of transport type for probability measures over \(\mathbb {R}^d\) . Since transportation of mass over \(\mathbb {R}^d\)  can be carried without limit, questions of boundary conditions do not arise there, unlike in the more recent works [9]}, [10]}, [11]} already cited above in which probability measures over \(\mathbb {R}_{\ge 0}^d\)  or the space of non-negative definite matrices are considered. We are not aware of previous works considering equations that involve derivatives of “affine” type, as we do here. In this context, the natural “movements” are different from those appearing for the transport geometry, and our additional constraint that we must deal with non-negative measures is the source of the necessity to address boundary issues.
